# Properties

 Label 1344.4.a.bj Level $1344$ Weight $4$ Character orbit 1344.a Self dual yes Analytic conductor $79.299$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1344 = 2^{6} \cdot 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1344.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$79.2985670477$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ Defining polynomial: $$x^{2} - x - 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 672) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{17}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -3 q^{3} + ( 5 - \beta ) q^{5} + 7 q^{7} + 9 q^{9} +O(q^{10})$$ $$q -3 q^{3} + ( 5 - \beta ) q^{5} + 7 q^{7} + 9 q^{9} + ( 11 + 11 \beta ) q^{11} + ( 12 - 2 \beta ) q^{13} + ( -15 + 3 \beta ) q^{15} + ( -75 + 11 \beta ) q^{17} + ( -4 - 32 \beta ) q^{19} -21 q^{21} + ( -41 - 33 \beta ) q^{23} + ( -83 - 10 \beta ) q^{25} -27 q^{27} + ( 18 + 32 \beta ) q^{29} + ( 44 - 20 \beta ) q^{31} + ( -33 - 33 \beta ) q^{33} + ( 35 - 7 \beta ) q^{35} + ( -144 + 14 \beta ) q^{37} + ( -36 + 6 \beta ) q^{39} + ( -193 + 61 \beta ) q^{41} + ( 172 + 76 \beta ) q^{43} + ( 45 - 9 \beta ) q^{45} + ( -138 + 110 \beta ) q^{47} + 49 q^{49} + ( 225 - 33 \beta ) q^{51} + ( -80 - 18 \beta ) q^{53} + ( -132 + 44 \beta ) q^{55} + ( 12 + 96 \beta ) q^{57} + ( 538 + 46 \beta ) q^{59} + ( -78 - 148 \beta ) q^{61} + 63 q^{63} + ( 94 - 22 \beta ) q^{65} + ( 686 + 6 \beta ) q^{67} + ( 123 + 99 \beta ) q^{69} + ( -551 - 55 \beta ) q^{71} + ( -120 - 178 \beta ) q^{73} + ( 249 + 30 \beta ) q^{75} + ( 77 + 77 \beta ) q^{77} + ( -206 + 106 \beta ) q^{79} + 81 q^{81} + ( 232 - 164 \beta ) q^{83} + ( -562 + 130 \beta ) q^{85} + ( -54 - 96 \beta ) q^{87} + ( -873 - 51 \beta ) q^{89} + ( 84 - 14 \beta ) q^{91} + ( -132 + 60 \beta ) q^{93} + ( 524 - 156 \beta ) q^{95} + ( 428 + 50 \beta ) q^{97} + ( 99 + 99 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 6q^{3} + 10q^{5} + 14q^{7} + 18q^{9} + O(q^{10})$$ $$2q - 6q^{3} + 10q^{5} + 14q^{7} + 18q^{9} + 22q^{11} + 24q^{13} - 30q^{15} - 150q^{17} - 8q^{19} - 42q^{21} - 82q^{23} - 166q^{25} - 54q^{27} + 36q^{29} + 88q^{31} - 66q^{33} + 70q^{35} - 288q^{37} - 72q^{39} - 386q^{41} + 344q^{43} + 90q^{45} - 276q^{47} + 98q^{49} + 450q^{51} - 160q^{53} - 264q^{55} + 24q^{57} + 1076q^{59} - 156q^{61} + 126q^{63} + 188q^{65} + 1372q^{67} + 246q^{69} - 1102q^{71} - 240q^{73} + 498q^{75} + 154q^{77} - 412q^{79} + 162q^{81} + 464q^{83} - 1124q^{85} - 108q^{87} - 1746q^{89} + 168q^{91} - 264q^{93} + 1048q^{95} + 856q^{97} + 198q^{99} + O(q^{100})$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 2.56155 −1.56155
0 −3.00000 0 0.876894 0 7.00000 0 9.00000 0
1.2 0 −3.00000 0 9.12311 0 7.00000 0 9.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$3$$ $$1$$
$$7$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1344.4.a.bj 2
4.b odd 2 1 1344.4.a.br 2
8.b even 2 1 672.4.a.j yes 2
8.d odd 2 1 672.4.a.e 2
24.f even 2 1 2016.4.a.q 2
24.h odd 2 1 2016.4.a.r 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
672.4.a.e 2 8.d odd 2 1
672.4.a.j yes 2 8.b even 2 1
1344.4.a.bj 2 1.a even 1 1 trivial
1344.4.a.br 2 4.b odd 2 1
2016.4.a.q 2 24.f even 2 1
2016.4.a.r 2 24.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1344))$$:

 $$T_{5}^{2} - 10 T_{5} + 8$$ $$T_{11}^{2} - 22 T_{11} - 1936$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$( 3 + T )^{2}$$
$5$ $$8 - 10 T + T^{2}$$
$7$ $$( -7 + T )^{2}$$
$11$ $$-1936 - 22 T + T^{2}$$
$13$ $$76 - 24 T + T^{2}$$
$17$ $$3568 + 150 T + T^{2}$$
$19$ $$-17392 + 8 T + T^{2}$$
$23$ $$-16832 + 82 T + T^{2}$$
$29$ $$-17084 - 36 T + T^{2}$$
$31$ $$-4864 - 88 T + T^{2}$$
$37$ $$17404 + 288 T + T^{2}$$
$41$ $$-26008 + 386 T + T^{2}$$
$43$ $$-68608 - 344 T + T^{2}$$
$47$ $$-186656 + 276 T + T^{2}$$
$53$ $$892 + 160 T + T^{2}$$
$59$ $$253472 - 1076 T + T^{2}$$
$61$ $$-366284 + 156 T + T^{2}$$
$67$ $$469984 - 1372 T + T^{2}$$
$71$ $$252176 + 1102 T + T^{2}$$
$73$ $$-524228 + 240 T + T^{2}$$
$79$ $$-148576 + 412 T + T^{2}$$
$83$ $$-403408 - 464 T + T^{2}$$
$89$ $$717912 + 1746 T + T^{2}$$
$97$ $$140684 - 856 T + T^{2}$$